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Hard, Math

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Alice and Bob are playing a game with a string of characters, with Alice going first. The string consists n characters, each of which is one of the first k letters of the alphabet. On a player’s turn, they can either arbitrarily permute the characters in the words, or delete exactly one character in the word (if there is at least one character). In addition, their resulting word cannot have appeared before throughout the entire game. The player unable to make a valid move loses the game.

Given \(n, k, p\), find the number of words with exactly n characters consisting of the first k letters of the alphabet such that Alice will win if both Alice and Bob play optimally. Return this number modulo the prime number p.

Input Format

The first line of input will contain three integers \(n, k, p\).

Output Format

Print a single integer, the number of winning words for Alice, modulo p.

Constraints

\(1 \leq n \leq 250\,000\)

\(1 \leq k \leq 26\)

\(10^8 \leq p \leq 10^9 + 100\)

p will be prime

SAMPLE INPUT
4 2 100000007
SAMPLE OUTPUT
14
Explanation

There are 14 strings that that Alice can win with. For example, some strings are "bbaa" and "baaa". Alice will lose on strings like "aaaa" or "bbbb".

Time Limit: 2.0 sec(s) for each input file.
Memory Limit: 256 MB
Source Limit: 1024 KB
Marking Scheme: Marks are awarded when all the testcases pass.
Allowed Languages: Bash, C, C++, C++14, Clojure, C#, D, Erlang, F#, Go, Groovy, Haskell, Java, Java 8, JavaScript(Rhino), JavaScript(Node.js), TypeScript, Julia, Kotlin, Lisp, Lisp (SBCL), Lua, Objective-C, OCaml, Octave, Pascal, Perl, PHP, Python, Python 3, R(RScript), Racket, Ruby, Rust, Scala, Swift, Swift-4.1, Visual Basic

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HackerEarth

Challenge Name

IndiaHacks 2017: Programming [Wave 2.0- Eliminator]

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